Given a probability space $(\Omega, \mathfrak S, P)$, a measurable function $f(x,y)$ and two random variables $X, Y : \mathbb \Omega \to \mathbb R$, then define $Z := f(X,Y)$. Then $Z$ is $\sigma(X,Y)$-measurable, i.e. $\sigma(Z) \subseteq \sigma(X,Y)$. But is there any finer way to describe $\sigma(Z)$?
For example, if $f$ just depends on the first coordinate, then $\sigma(Z) \subseteq \sigma(X)$, i.e. it is enough to know $X$ to determine $Z$. If we have $x, y, y'$ such that $f(x,y) \ne f(x,y')$ and $X(\omega) = x, Y(\omega) = y, X(\omega') = x, Y(\omega') = y$ and $c := f(x,y)$ then $\{ Z = c \} \ne \{ X = x \}$ but $\{ Z = c \} \cap \{ X = x \} \ne \emptyset$, and as for $A \in \sigma(X)$, either $A \cap \{ X = x \} = \emptyset$ or $\{ X = x \}\subseteq A$ we have $\{ Z = c \} \notin \sigma(X)$. If $f$ is bijective and has a measurable inverse, then $\sigma(Z) = \sigma(X,Y)$. For if $\{ X \in A, Y \in B \}$ with $A, B \in \mathfrak B(\mathbb R)$, then $C := f(A, B) \in \mathfrak B(\mathbb R)$ and we have $\{ Z \in C \} = \{ X \in A, Y \in B \}$. As if $Z(\omega) \in C$, then $Z(\omega) = f(a,b)$ with $a \in A, b \in B$ and $(X(\omega), Y(\omega)) = f^{-1}(f(a,b)) = (a,b)$, hence $(X(\omega), Y(\omega)) \in A \times B$, and conversely if $(X(\omega), Y(\omega)) = (a,b)$ with $(a,b) \in A\times B$ then $Z(\omega) = f(a,b) \in C$.
These are some cases, but is there a finer characterization of $\sigma(Z)$ dependent on $f : \mathbb R^2 \to \mathbb R$?